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User blog:Hl3 or bust/A Newcomer Tries His Hand At a Thing
Welcome to my first attempt to do anything here other than make like 2 edits to the newest croutonillion page and lurk, i hope you find it lacking so i can get the motivation to do something actually good: Letter Notation part 1: A(a) = a+1 A1(a)/A1(a) = A(a) An(a) indicates an iterated function (i.e. A(A(A(....A(A(a))....))) with a A's) An(a) = An-1An-1An-1....An-1a(a....(a)(a)(a) with a A's &(a) where $ is the nth letter* = @@@....@a(a)....(a)(a)(a) with a @'s, where @ is the n-1th letter *due to the large numbers created by this, we need to create a hypothetical nth letter where n>26, which we can just use & for right now Analysis: A(a) = F0(a) is the FGH Aa(a) = F1(a) A2(a) = F2(a) in general, Aa(a) ≈ Fw(a) (where w is an ASCII substitute for omega) B(a) = AAA....Aa(a)....(a)(a)(a) with a A's ≈ Fw+1(a) Ba(a) ≈ Fw+2(a) B2(a) ≈ Fw+3(a) Ba(a) ≈ Fw+(n+1)(a) C(a) = BBB....Ba(a)....(a)(a)(a) ≈ Fw2+1(a) D(a) ≈ Fw3+1(a) &(a) where & is the ath letter ≈ Fw(n-1)+1(a) with this, we see that Letter Notation part 1 (LNP1), via the function F(n) = &(n) where & is the nth letter, has a growth rate of w2, which isn't particulary great (roughly equal to {a,a,a,a} in BEAF), but it's a start Letter Notation part 2 (LNP2): $AAA.....AAA(a) where $ is the any letter other than A = @&&&....&&&(a) where & is the a-1th letter $$$....$$$(a) where the first $ is any letter other than A = $$$....$$&$$$....$$&$$$....$$&....$$$....$$&a(a)....(a)(a)(a) with a $$$....$$&'s where & is the letter preceeding $ (may be A) $1(a) = $$$....$$$(a) with a $'s $b(a) = $b-1b-1....b-1b-1(a) with a b-1's $$$...$$$bc....xy(a) where the first $ is any letter other than A = $$$...$$&bc....xy$$$...$$&bc....xy$$$...$$&bc....xy....$$$...$$&bc....xya(a)....(a)(a)(a) with a $$$...$$&bc....xy's where & is the letter preceeding $ with this, AA is a synonym of &, or the ath letter, since the A on the right becomes & and the other A is the "0th" letter, which isn't a thing analysis (may be slightly off as i'm writing this while mostly asleep): AA(a) ≈ Fw2(a) AAa(a) ≈ Fw2+1(a) AA2(a) ≈ Fw2+2(a) AAa(a) ≈ Fw2+w(a) (could also be w22, but i can't remember if that breaks down to w2+w2 or not) AB(a) ≈ Fw2+w+1(a) A&(a) where & is the ath letter ≈ Fw2+w(a-1)+1(a) BA(a) ≈ Fw2+w2(a) &A(a) ≈ Fw3(a) (roughtly {a,a,a,a,a} in BEAF) AAA(a) = &&(a) ≈ Fw3+w2(a) A1(a) ≈ Fww(a) (roughly {a,a+2(1)2} in BEAF) A2(a) = A111....11(a) with a 1's ≈ Fww+1(a) (very rough approximation) Aa(a) ≈ Fww2(a) via a function F(a) = Aa(a), we see that LNP2 (or at least the part with a's) has a growth rate of ww2, which is a really big jump over what LNP1 could make LNP3: with the issue of the previous parts mostly cleared up, it's time to extend this A = a &[....[[[[1]]....]]]{a) with n []'s = &[....[[&[1]....]]](a) with n-1 []'s AAA.....AAA with n A's = A*A*A.....*A*A*A with n A's = An A*A*A....A*A*a = (A*A*A....A*A*(a-1))+a (@)+n (where @ is any string of &'s, optionally with an a or n at the end) is counted as the next "letter" after (@)+(n-1) (@)+0 = (@) &[....[[[[n]]....]]](a) with b []'s = &[[[....[[&n-1n-1n-1....n-1n-1]]....]]](a) with a n-1's and b-1 []'s these form new rules form an FGH type structure, but instead of w's, it's A's analysis: AA(a) ≈ ww2 Aaa(a) ≈ ww2+1 Aaa ≈ ww2+w AA+1(a) ≈ ww2+w+1 AA*2(a) ≈ ww2+w2 A[[1]](a) = AAA(a) ≈ ww2+ww A[1](a) = A[[A]](a) = A[[a]](a) ≈ ww2+1 A[....[[[[1]]....]]](a) with a []'s ≈ ww2 this hasn't gotten us very far it seems, but we can extend it further []'s are 1st level separators, as they break down into &'s, and we can call {}'s 2nd level separators since they will break down into 1st level separators, and in general: A%1%(a) where % is an nth level separator = &$$$....$$&$$....$$$(a) with a $'s, where $ is an n-1th level separator so A{1}(a) = A[....[[[A]....]]](a) with a []'s ≈ ww2+ww2 in general: A@1@(a) where @ is an ath level separator ≈ www Category:Blog posts